Strong rational connectedness of surfaces
type: Item
instance of: scholarly article
title: Strong rational connectedness of surfaces
zbMATH Open document ID: 1246.14064
DOI: 10.1515/CRELLE.2011.108
publication date: 2012-05-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/0810.2597
Mathematics Subject Classification ID: 14M22
Mathematics Subject Classification ID: 14J26
zbMATH DE Number: 6034177
zbMATH Keywords: strong rational connectedness
zbMATH Keywords: log del Pezzo surface
author: Chenyang Xu
MaRDI profile type: Publication
arXiv ID: 0810.2597
cites work: Compactifying the space of stable maps
cites work: Versal deformations and algebraic stacks
cites work: The irreducibility of the space of curves of a given genus
cites work: Families of rationally connected varieties
cites work: On Shokurov's rational connectedness conjecture
cites work: Nonrational Hypersurfaces
cites work: Deformation theory of representable morphisms of algebraic stacks
cites work: \(\underline{\Hom}\)-stacks and restriction of scalars
cites work: Intersection theory on algebraic stacks and on their moduli spaces
OpenAlex ID: W2110120279
published in: Journal für die Reine und Angewandte Mathematik
review text: Let \(X\) be a complex projective variety, then \(X\) is rationally connected if for any two very general points \(x,y\in X\) there is a rational curve \(f:\mathbb P ^1\to X\) such that \(x,y\in f(\mathbb P ^1)\). By a result of Campana and Kollár-Miyaoka-Mori, it is known that all smooth Fano varieties are rationally connected and by a result of Zhang, the same is true for log Fano varieties i.e. for projective klt pairs \((X,\Delta )\) such that \(X\) is normal and \(-(K_X+\Delta)\) is ample. It is also known that if \(X\) is smooth then \(X\) is rationally connected if and only if it is strongly rationally connected so that for any \(x\in X\) there is a rational curve \(f:\mathbb P ^1\to X\) such that \(x\in f(\mathbb P ^1)\) and \(f^*T_X\) is ample. One can also ask if the smooth locus of a log Fano variety is rationally connected. In general this is a very hard question, which in dimension \(2\) was affirmatively answered by Keel and M{c}Kernan. In the paper under review it is shown that the smooth locus of a two dimensional log Fano variety is strongly rationally connected and that if \(S\) is a projective surface with at worst Du Val singularities such that the smooth locus \(S^{\text{sm}}\) is rationally connected, then \(S^{\text{sm}}\) is strongly rationally connected.
reviewed by: Christopher Hacon
description: scientific article; zbMATH DE number 6034177